I am looking for a proof of one of the integrals presented in Harry Bateman's *Tables of Integral Transforms*. The specific integral in question is presented on page 42 in chapter 8.9 as equation (3):

$$ \int_0^{\infty} \left[x^{\nu+\frac{1}{2}} \mathrm{e}^{-\frac{1}{2} x^2} L^{\nu}_n\!\left(x^2\right) \right]J_{\nu}\!\left(xy\right) \sqrt{xy}\,\mathrm{d}x = \left(-1\right)^n \mathrm{e}^{-\frac{1}{2} y^2} y^{\nu+\frac{1}{2}} L^{\nu}_n\!\left(y^2\right) $$

where $J_{\nu}$ are the Bessel functions of the first kind and $L^{\nu}_n$ are the associated Laguerre polynomials. This integral is basically the Hankel transform of the function $f\!\left(x\right) = x^{\nu} \mathrm{e}^{-\frac{1}{2} x^2} L^{\nu}_n\!\left(x^2\right) $ with an additional factor $\sqrt{y}$ (depending on the definition).

A similar integral also appears in this question, but the integral in question here is a bit different.

The full reference of the table entry is:

Harry Bateman, *Tables of Integral Transforms*, Volume 2, p. 42 (ch. 8.9, eq. (3))